class: center, middle, inverse, title-slide .title[ # Statistical Concepts Everyone Should Know ] .subtitle[ ## <p><em>Statistics for Life</em></p> ] .author[ ### <p><strong>John Slough</strong><br /> <em>for</em> <strong>The John & Calvin Podcast</strong></p> ] --- class: inverse, center, large <h1 style="font-size: 80px; margin-top: 200px;">Beware the Average</h1> <hr style="margin-top: 2em; margin-bottom: 2em;"> -- <h2 style="font-size: 60px;">It's Meaner than the Median</h2> --- ## 1. Mean vs Median ### Local bar weath distribution: 50 Patrons <div id="htmlwidget-983315589fed18d622a7" style="width:750px;height:450px;" class="plotly html-widget"></div> <script type="application/json" 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style='color:#C04E3D'><b>Mean:<\/b><\/span> $64,990<br><span style='color:#BFA06D'><b>Median:<\/b><\/span> $58,160","showarrow":false,"align":"right","font":{"size":18,"color":"black"}}]},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"32111c359b57":{"x":{},"type":"bar"},"32113697b39":{"xintercept":{}},"3211218dcef":{"xintercept":{}}},"cur_data":"32111c359b57","visdat":{"32111c359b57":["function (y) ","x"],"32113697b39":["function (y) ","x"],"3211218dcef":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- ## 1. Mean vs Median ### Local bar weath distribution: 50 Patrons + Bill Gates <div id="htmlwidget-6f1d8b9e786d9d3fe28d" style="width:750px;height:450px;" class="plotly html-widget"></div> <script type="application/json" 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style='color:#C04E3D'><b>Mean:<\/b><\/span> $2,941,240,186<br><span style='color:#BFA06D'><b>Median:<\/b><\/span> $58,410","showarrow":false,"align":"right","font":{"size":18,"color":"black"}}]},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"32111d6c1043":{"x":{},"type":"bar"},"3211219fadcc":{"xintercept":{}}},"cur_data":"32111d6c1043","visdat":{"32111d6c1043":["function (y) ","x"],"3211219fadcc":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- ## 1. Mean vs Median <div style="font-size: 130%; text-align: left; margin: 100px 0;"> <p><strong>Mean</strong>: Sum of all values divided by the number of values.</p> <br> <p><strong>Median</strong>: Middle value when the values are ordered.</p> </div> --- ## 1. Mean vs Median ### Impact of a $150B Outlier on Mean vs. Median — Across Sample Sizes <table class="table" style="font-size: 24px; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:right;"> Sample Size </th> <th style="text-align:right;"> Mean </th> <th style="text-align:right;"> Median </th> </tr> </thead> <tbody> <tr> <td style="text-align:right;"> 101 </td> <td style="text-align:right;padding-left: 20px;"> $1,485,210,710 </td> <td style="text-align:right;padding-left: 20px;"> $62,647 </td> </tr> <tr> <td style="text-align:right;"> 1,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 149,914,454 </td> <td style="text-align:right;padding-left: 20px;"> $60,030 </td> </tr> <tr> <td style="text-align:right;"> 10,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 15,062,963 </td> <td style="text-align:right;padding-left: 20px;"> $59,642 </td> </tr> <tr> <td style="text-align:right;"> 100,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 1,564,871 </td> <td style="text-align:right;padding-left: 20px;"> $59,868 </td> </tr> <tr> <td style="text-align:right;"> 1,000,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 214,870 </td> <td style="text-align:right;padding-left: 20px;"> $59,913 </td> </tr> <tr> <td style="text-align:right;"> 10,000,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 79,865 </td> <td style="text-align:right;padding-left: 20px;"> $59,877 </td> </tr> <tr> <td style="text-align:right;"> 100,000,001 </td> <td style="text-align:right;padding-left: 20px;"> $ 66,366 </td> <td style="text-align:right;padding-left: 20px;"> $59,877 </td> </tr> </tbody> </table> --- ## 1. Mean vs Median A more realistic example  --- .pull-left[ ### Economics & Wealth - GDP per capita - household income - net worth - home price - monthly rent - CEO compensation ### Health & Healthcare - life expectancy - healthcare spending per person - hospital bill - patient out-of-pocket cost ] .pull-right[ ### Academics - test score (SAT, PISA) - GPA - academic citations - speaking invitations per expert ### Other - commute time - screen-time per user - household energy use - carbon emissions per capita - YouTube ad revenue per channel - revenue per app in the App Store - Software Bug Fix Times ] --- ## Mean or Median? &mdash; It Depends on the Shape of Your Data - **Symmetric (bell-shaped) distributions** *Mean ≈ Median* → either works <span style="font-size:0.9em;color:gray;">(heights, random errors, test scores in large classes)</span> - **Skewed distributions (long tail on one side)** **Use the Median** &nbsp;– it ignores extreme outliers <span style="font-size:0.9em;color:gray;">(wealth, medical bills, wait times)</span> - **Heavy-tailed or “black-swan” data** Median or percentiles are safest; mean can explode <span style="font-size:0.9em;color:gray;">(earthquake energy, insurance losses)</span> - **Bimodal / multi-cluster data** No single centre makes sense → show the full distribution or the two modes <span style="font-size:0.9em;color:gray;">(commuter travel times, exam grades with pass/fail peaks)</span> .small[ *Rule of thumb ▶ If a handful of extreme observations could swing the result, report the **median** (and a spread measure) instead of the mean.* ] --- ### From “What’s the Average?” → “What’s the Distribution?” - **Why the mean mis-led us:** the income data are **skewed**, so a few extreme values dragged the mean away from the typical person (≈ median). - **What really matters is the full shape** of the data—its **distribution**. - **Normal distribution (bell curve)** - Symmetric, mean = median = mode. - Central Limit Theorem ⇒ sample means often look normal. - **But many real-world data aren’t normal:** - Incomes/wealth → **log-normal / Pareto** (long right tail) - Failure times → **Weibull / Exponential** (many early, few late) - Web traffic, social-media posts, bug lifetimes … often heavy-tailed .small[ **Takeaway ▶** Before quoting a single “average,” look at the distribution. If it’s skewed or heavy-tailed, report the **median or percentiles** and pick the distribution that actually fits the data. ] --- ## Distributions ### Types and Skew --- ## Bias **Bias** is the difference between the expected value of an estimator and the true value of the parameter it estimates. Formally: $$ \text{Bias}(\hat{\theta}) = \mathbb{E}[\hat{\theta}] - \theta $$ Where: - `\(\hat{\theta}\)` = the **estimator** (your calculated estimate) - `\(\theta\)` = the **true parameter** (the real value you want) - `\(\mathbb{E}[\hat{\theta}]\)` = the **expected value** of the estimator (its long-run average over many samples) - If Bias = 0 → the estimator is **unbiased**. - If Bias ≠ 0 → the estimator is **biased** (systematically too high or too low). --- Bias doesn’t just shift your first guess. It filters what you see next — making you even more biased over time Selection bias: The sample is not representative of the population. Omitted variable bias: Leaving out a variable that influences both the dependent and independent variables. Measurement bias (or information bias): Errors in how data is collected or recorded. Survivorship bias: Only analyzing "survivors" or those who remain, ignoring those who dropped out or failed. Recall bias: Errors because people remember things inaccurately (common in surveys and retrospective studies). Observer bias: Researcher's expectations subtly influence measurements or observations. Publication bias: Studies with "positive" results are more likely to be published than "null" or "negative" results. --- ## Bias ### The Self-Reinforcing Feedback Loop of Bias <div class="grViz html-widget html-fill-item" id="htmlwidget-dcdfc64296f5794e54a4" style="width:504px;height:504px;"></div> <script type="application/json" data-for="htmlwidget-dcdfc64296f5794e54a4">{"x":{"diagram":"\ndigraph bias_loop {\n\n graph [layout = circo]\n node [shape = box, style = \"rounded,filled\", fontname = Helvetica, fontsize=12]\n\n # Node Colors\n CurrentBias [label = \"Current Bias\", fillcolor = \"#F5C4BB\"] # lightest red\n SelectiveAttention [label = \"Selective Attention\", fillcolor = \"#EDA89C\"] # medium-light\n SelectiveEvidence [label = \"Selective Evidence\", fillcolor = \"#E17A66\"] # medium\n ReinforcedBias [label = \"Reinforced Bias\", fillcolor = \"#C04E3D\"] # strong bias (your mean_color)\n\n edge [penwidth = 2]\n\n CurrentBias -> SelectiveAttention\n SelectiveAttention -> 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